# Apparent motions of the Sun and Moon

Today, we'll look at the apparent motions of the Sun and Moon through the Earth's sky.

#### The Sun's path through the sky: seasons

Here in Rochester, New York, there are four very distinct seasons throughout the year: spring, summer, autumn and winter. Each has its own typical temperature and length of day and night.

```
Q:  What causes the seasons?

```

The answer is NOT "the distance between the Earth and the Sun." In fact, the Earth's slightly elliptical orbit brings it a bit closest to the Sun in January: right in the middle of the coldest time of the year for us in Rochester!

```
Month         Distance from Sun
-----------------------------------------
January         0.9833  AU

April           0.9996

July            1.0151

October         0.9984
-----------------------------------------
```
The table above expresses distances in terms of the Astronomical Unit (AU for short), which is defined as the semimajor axis of the Earth's orbit:
• 1 AU = 149597870.691 km (approx 150 million km)

No, it's not the distance between Sun and Earth that causes the seasons: instead, it's the angle at which sunlight strikes the Earth's surface. Suppose that the Earth's rotation axis were perpendicular to the plane of its orbit around the Sun, like this:

No matter where the Earth might be in its orbit, at noon, sunlight would strike the ground in Rochester at the same angle.

```
Q:  The latitude of Rochester is about +43 degrees North.
At what angle away from the normal
(i.e. how far from perpendicular)
would sunlight strike the ground at noon?

```

However, in real life, the Earth's rotation axis is NOT perpendicular to the plane of its orbit. Instead, the rotation axis is tilted by about 23.5 degrees away from the perpendicular. That means that when the Earth is on one side of the Sun in its orbit, in June, sunlight strikes the ground most directly: we call this the summer solstice.

Six months later, in January, when the Earth is on the other side of its orbit, sunlight strikes the northern hemisphere with a more glancing blow: we call this the winter solstice.

```
Q:  Recall that the latitude of Rochester is
about +43 degrees North.  If it is high summer
in Rochester, at what angle from the perpendicular
does sunlight strike the Earth at noon?

Q:  At what angle does sunlight strike the Earth
at noon during the depths of the Rochester winter?

```

Maybe it would help to see the changing direction of the Sun's rays on the Earth -- from space! Click on the picture to activate.

You can see the difference in the Sun's path through our sky over the course of the seasons in one of Regina Valkenborgh's solargraphs, shown below.

#### Phases of the Moon

As the Moon orbits around the Earth, we see different phases, ranging from (the invisible) New Moon through First Quarter, Full Moon, Third Quarter, and back to New Moon.

There are several different periods which astronomers use to describe the motion of the Moon.

• sidereal period is the time it takes for the Moon to move by 360 degrees around the Earth, as seen by an observer floating high above the plane of the solar system. The sidereal period of the Moon's orbit is about 27.3 days.

• synodic period is the time from one Full Moon to the next Full Moon (or from one New Moon to the next New Moon). This is a little longer than the sidereal period because the Moon has to "catch up" to the motion of the Earth around the Sun. The synodic period of the Moon's orbit is about 29.5 days.

It turns out that the synodic and sidereal periods of a moon around its planet are related to the planet's orbital period around the Sun:

```

1                        1                       1
---------------   =    ---------------  -  -----------------------
synodic period         sidereal period     planet's orbital period

```

If you look more closely at the Moon over the course of a full cycle, you'll start to see small variations in its size -- due to the eccentricity of the Moon's orbit around the Earth -- and very small changes in the visibility of regions near the limbs -- again due to the eccentricity of the Moon's orbit.

Thanks to Astronomy Picture of the Day and Antonio Cidadao

#### Eclipses

We on Earth are lucky to be the beneficiaries of a cosmic coincidence (and I really mean that!). Consider the apparent angular size of the Sun, as seen from the Earth:

We can calculate the angle theta using trigonometry:

```
Q:  Given diameter of Sun     d = 1.39 million km,
and distance to the Sun   L = 150 million km,

what is the apparent angular size of the Sun?

Q:  Given diameter of Moon    d = 3476 km,
and distance to the Moon  L = 384,000 km,

what is the apparent angular size of the Moon?

```

As a result of this coincidence, we see a beautiful phenomenon -- the solar eclipse when the Moon passes between the Sun and Earth:

We also see lunar eclipses when the Moon passes through the Earth's shadow:

```
Q:  Is there a solar eclipse during every New Moon?

```

#### Homework

1. Sunlight carries quite a bit of energy, even at the Earth's distance from the Sun. Suppose we purchase a solar power panel, one meter on a side, with an efficiency of 100% (real commercial solar panels have an efficiency of only about 10%). If sunlight strikes our panel face-on, it provides about 1340 Watts (that's 1340 Joules of energy each second). However, if sunlight strikes the panel at an angle, it provides less power. You can calculate the exact amount by finding the "effective area" of the panel: it depends on the cosine of the angle between the sunlight and the normal to the panel.

So, we place our panel flat on the ground in Rochester, New York.

1. What is the power produced by the panel at noon on the summer solstice?
2. What is the power produced by the panel at noon on the winter solstice?
3. What is the power produced by the panel at noon on the equinox?

2. The Sun, Earth and Moon are unusual in having just the right combination of orbital distances and sizes to cause nearly total solar eclipses: the Moon is almost exactly the same apparent angular size as the Sun, as seen from the Earth. Are there any other moons in the solar system which, when viewed from their home planets, are also roughly the same apparent angular size as the Sun? Find the best planet-moon combination you can. Extra points for a list of the best three combinations. You may find The Nine Planets web site a good resource.